Internet Engineering Task Force (IETF) K. Moriarty, Ed.
Request for Comments: 8017 EMC Corporation
Obsoletes: 3447 B. Kaliski
Category: Informational Verisign
ISSN: 2070-1721 J. Jonsson
Subset AB
A. Rusch
RSA
November 2016
PKCS #1: RSA Cryptography Specifications Version 2.2
Abstract
This document provides recommendations for the implementation of
public-key cryptography based on the RSA algorithm, covering
cryptographic primitives, encryption schemes, signature schemes with
appendix, and ASN.1 syntax for representing keys and for identifying
the schemes.
This document represents a republication of PKCS #1 v2.2 from RSA
Laboratories' Public-Key Cryptography Standards (PKCS) series. By
publishing this RFC, change control is transferred to the IETF.
This document also obsoletes RFC 3447.
Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This document is a product of the Internet Engineering Task Force
(IETF). It represents the consensus of the IETF community. It has
received public review and has been approved for publication by the
Internet Engineering Steering Group (IESG). Not all documents
approved by the IESG are a candidate for any level of Internet
Standard; see Section 2 of RFC 7841.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
http://www.rfc-editor.org/info/rfc8017.
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Copyright Notice
Copyright (c) 2016 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(http://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document. Code Components extracted from this document must
include Simplified BSD License text as described in Section 4.e of
the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
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0x indicator of hexadecimal representation of an octet
or an octet string: "0x48" denotes the octet with
hexadecimal value 48; "(0x)48 09 0e" denotes the
string of three consecutive octets with hexadecimal
values 48, 09, and 0e, respectively
\lambda(n) LCM(r_1-1, r_2-1, ... , r_u-1)
\xor bit-wise exclusive-or of two octet strings
\ceil(.) ceiling function; \ceil(x) is the smallest integer
larger than or equal to the real number x
|| concatenation operator
== congruence symbol; a == b (mod n) means that the
integer n divides the integer a - b
Note: The Chinese Remainder Theorem (CRT) can be applied in a non-
recursive as well as a recursive way. In this document, a recursive
approach following Garner's algorithm [GARNER] is used. See also
Note 1 in Section 3.2.
3. Key Types
Two key types are employed in the primitives and schemes defined in
this document: RSA public key and RSA private key. Together, an RSA
public key and an RSA private key form an RSA key pair.
This specification supports so-called "multi-prime" RSA where the
modulus may have more than two prime factors. The benefit of multi-
prime RSA is lower computational cost for the decryption and
signature primitives, provided that the CRT is used. Better
performance can be achieved on single processor platforms, but to a
greater extent on multiprocessor platforms, where the modular
exponentiations involved can be done in parallel.
For a discussion on how multi-prime affects the security of the RSA
cryptosystem, the reader is referred to [SILVERMAN].
3.1. RSA Public Key
For the purposes of this document, an RSA public key consists of two
components:
n the RSA modulus, a positive integer
e the RSA public exponent, a positive integer
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In a valid RSA public key, the RSA modulus n is a product of u
distinct odd primes r_i, i = 1, 2, ..., u, where u >= 2, and the RSA
public exponent e is an integer between 3 and n - 1 satisfying
GCD(e,\lambda(n)) = 1, where \lambda(n) = LCM(r_1 - 1, ..., r_u - 1).
By convention, the first two primes r_1 and r_2 may also be denoted p
and q, respectively.
A recommended syntax for interchanging RSA public keys between
implementations is given in Appendix A.1.1; an implementation's
internal representation may differ.
3.2. RSA Private Key
For the purposes of this document, an RSA private key may have either
of two representations.
1. The first representation consists of the pair (n, d), where the
components have the following meanings:
n the RSA modulus, a positive integer
d the RSA private exponent, a positive integer
2. The second representation consists of a quintuple (p, q, dP, dQ,
qInv) and a (possibly empty) sequence of triplets (r_i, d_i,
t_i), i = 3, ..., u, one for each prime not in the quintuple,
where the components have the following meanings:
p the first factor, a positive integer
q the second factor, a positive integer
dP the first factor's CRT exponent, a positive integer
dQ the second factor's CRT exponent, a positive integer
qInv the (first) CRT coefficient, a positive integer
r_i the i-th factor, a positive integer
d_i the i-th factor's CRT exponent, a positive integer
t_i the i-th factor's CRT coefficient, a positive integer
In a valid RSA private key with the first representation, the RSA
modulus n is the same as in the corresponding RSA public key and is
the product of u distinct odd primes r_i, i = 1, 2, ..., u, where u
>= 2. The RSA private exponent d is a positive integer less than n
satisfying
e * d == 1 (mod \lambda(n)),
where e is the corresponding RSA public exponent and \lambda(n) is
defined as in Section 3.1.
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In a valid RSA private key with the second representation, the two
factors p and q are the first two prime factors of the RSA modulus n
(i.e., r_1 and r_2); the CRT exponents dP and dQ are positive
integers less than p and q, respectively, satisfying
e * dP == 1 (mod (p-1))
e * dQ == 1 (mod (q-1)) ,
and the CRT coefficient qInv is a positive integer less than p
satisfying
q * qInv == 1 (mod p).
If u > 2, the representation will include one or more triplets (r_i,
d_i, t_i), i = 3, ..., u. The factors r_i are the additional prime
factors of the RSA modulus n. Each CRT exponent d_i (i = 3, ..., u)
satisfies
e * d_i == 1 (mod (r_i - 1)).
Each CRT coefficient t_i (i = 3, ..., u) is a positive integer less
than r_i satisfying
R_i * t_i == 1 (mod r_i) ,
where R_i = r_1 * r_2 * ... * r_(i-1).
A recommended syntax for interchanging RSA private keys between
implementations, which includes components from both representations,
is given in Appendix A.1.2; an implementation's internal
representation may differ.
Notes:
1. The definition of the CRT coefficients here and the formulas that
use them in the primitives in Section 5 generally follow Garner's
algorithm [GARNER] (see also Algorithm 14.71 in [HANDBOOK]).
However, for compatibility with the representations of RSA
private keys in PKCS #1 v2.0 and previous versions, the roles of
p and q are reversed compared to the rest of the primes. Thus,
the first CRT coefficient, qInv, is defined as the inverse of q
mod p, rather than as the inverse of R_1 mod r_2, i.e., of
p mod q.
2. Quisquater and Couvreur [FASTDEC] observed the benefit of
applying the CRT to RSA operations.
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RFC 8017 PKCS #1 v2.2 November 20164. Data Conversion Primitives
Two data conversion primitives are employed in the schemes defined in
this document:
o I2OSP - Integer-to-Octet-String primitive
o OS2IP - Octet-String-to-Integer primitive
For the purposes of this document, and consistent with ASN.1 syntax,
an octet string is an ordered sequence of octets (eight-bit bytes).
The sequence is indexed from first (conventionally, leftmost) to last
(rightmost). For purposes of conversion to and from integers, the
first octet is considered the most significant in the following
conversion primitives.
4.1. I2OSP
I2OSP converts a nonnegative integer to an octet string of a
specified length.
I2OSP (x, xLen)
Input:
x nonnegative integer to be converted
xLen intended length of the resulting octet string
Output:
X corresponding octet string of length xLen
Error: "integer too large"
Steps:
1. If x >= 256^xLen, output "integer too large" and stop.
2. Write the integer x in its unique xLen-digit representation in
base 256:
x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) + ...
+ x_1 256 + x_0,
where 0 <= x_i < 256 (note that one or more leading digits
will be zero if x is less than 256^(xLen-1)).
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3. Let the octet X_i have the integer value x_(xLen-i) for 1 <= i
<= xLen. Output the octet string
X = X_1 X_2 ... X_xLen.
4.2. OS2IP
OS2IP converts an octet string to a nonnegative integer.
OS2IP (X)
Input: X octet string to be converted
Output: x corresponding nonnegative integer
Steps:
1. Let X_1 X_2 ... X_xLen be the octets of X from first to last,
and let x_(xLen-i) be the integer value of the octet X_i for 1
<= i <= xLen.
2. Let x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) +
... + x_1 256 + x_0.
3. Output x.
5. Cryptographic Primitives
Cryptographic primitives are basic mathematical operations on which
cryptographic schemes can be built. They are intended for
implementation in hardware or as software modules and are not
intended to provide security apart from a scheme.
Four types of primitive are specified in this document, organized in
pairs: encryption and decryption; and signature and verification.
The specifications of the primitives assume that certain conditions
are met by the inputs, in particular that RSA public and private keys
are valid.
5.1. Encryption and Decryption Primitives
An encryption primitive produces a ciphertext representative from a
message representative under the control of a public key, and a
decryption primitive recovers the message representative from the
ciphertext representative under the control of the corresponding
private key.
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One pair of encryption and decryption primitives is employed in the
encryption schemes defined in this document and is specified here:
RSA Encryption Primitive (RSAEP) / RSA Decryption Primitive (RSADP).
RSAEP and RSADP involve the same mathematical operation, with
different keys as input. The primitives defined here are the same as
Integer Factorization Encryption Primitive using RSA (IFEP-RSA) /
Integer Factorization Decryption Primitive using RSA (IFDP-RSA) in
IEEE 1363 [IEEE1363] (except that support for multi-prime RSA has
been added) and are compatible with PKCS #1 v1.5.
The main mathematical operation in each primitive is exponentiation.
5.1.1. RSAEP
RSAEP ((n, e), m)
Input:
(n, e) RSA public key
m message representative, an integer between 0 and n - 1
Output: c ciphertext representative, an integer between 0 and n - 1
Error: "message representative out of range"
Assumption: RSA public key (n, e) is valid
Steps:
1. If the message representative m is not between 0 and n - 1,
output "message representative out of range" and stop.
2. Let c = m^e mod n.
3. Output c.
5.1.2. RSADP
RSADP (K, c)
Input:
K RSA private key, where K has one of the following forms:
+ a pair (n, d)
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RFC 8017 PKCS #1 v2.2 November 20165.2. Signature and Verification Primitives
A signature primitive produces a signature representative from a
message representative under the control of a private key, and a
verification primitive recovers the message representative from the
signature representative under the control of the corresponding
public key. One pair of signature and verification primitives is
employed in the signature schemes defined in this document and is
specified here: RSA Signature Primitive, version 1 (RSASP1) / RSA
Verification Primitive, version 1 (RSAVP1).
The primitives defined here are the same as Integer Factorization
Signature Primitive using RSA, version 1 (IFSP-RSA1) / Integer
Factorization Verification Primitive using RSA, version 1 (IFVP-RSA1)
in IEEE 1363 [IEEE1363] (except that support for multi-prime RSA has
been added) and are compatible with PKCS #1 v1.5.
The main mathematical operation in each primitive is exponentiation,
as in the encryption and decryption primitives of Section 5.1.
RSASP1 and RSAVP1 are the same as RSADP and RSAEP except for the
names of their input and output arguments; they are distinguished as
they are intended for different purposes.
5.2.1. RSASP1
RSASP1 (K, m)
Input:
K RSA private key, where K has one of the following forms:
- a pair (n, d)
- a quintuple (p, q, dP, dQ, qInv) and a (possibly empty)
sequence of triplets (r_i, d_i, t_i), i = 3, ..., u
m message representative, an integer between 0 and n - 1
Output:
s signature representative, an integer between 0 and n - 1
Error: "message representative out of range"
Assumption: RSA private key K is valid
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Output:
m message representative, an integer between 0 and n - 1
Error: "signature representative out of range"
Assumption: RSA public key (n, e) is valid
Steps:
1. If the signature representative s is not between 0 and n - 1,
output "signature representative out of range" and stop.
2. Let m = s^e mod n.
3. Output m.
6. Overview of Schemes
A scheme combines cryptographic primitives and other techniques to
achieve a particular security goal. Two types of scheme are
specified in this document: encryption schemes and signature schemes
with appendix.
The schemes specified in this document are limited in scope in that
their operations consist only of steps to process data with an RSA
public or private key, and they do not include steps for obtaining or
validating the key. Thus, in addition to the scheme operations, an
application will typically include key management operations by which
parties may select RSA public and private keys for a scheme
operation. The specific additional operations and other details are
outside the scope of this document.
As was the case for the cryptographic primitives (Section 5), the
specifications of scheme operations assume that certain conditions
are met by the inputs, in particular that RSA public and private keys
are valid. The behavior of an implementation is thus unspecified
when a key is invalid. The impact of such unspecified behavior
depends on the application. Possible means of addressing key
validation include explicit key validation by the application; key
validation within the public-key infrastructure; and assignment of
liability for operations performed with an invalid key to the party
who generated the key.
A generally good cryptographic practice is to employ a given RSA key
pair in only one scheme. This avoids the risk that vulnerability in
one scheme may compromise the security of the other and may be
essential to maintain provable security. While RSAES-PKCS1-v1_5
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RFC 8017 PKCS #1 v2.2 November 2016
(Section 7.2) and RSASSA-PKCS1-v1_5 (Section 8.2) have traditionally
been employed together without any known bad interactions (indeed,
this is the model introduced by PKCS #1 v1.5), such a combined use of
an RSA key pair is NOT RECOMMENDED for new applications.
To illustrate the risks related to the employment of an RSA key pair
in more than one scheme, suppose an RSA key pair is employed in both
RSAES-OAEP (Section 7.1) and RSAES-PKCS1-v1_5. Although RSAES-OAEP
by itself would resist attack, an opponent might be able to exploit a
weakness in the implementation of RSAES-PKCS1-v1_5 to recover
messages encrypted with either scheme. As another example, suppose
an RSA key pair is employed in both RSASSA-PSS (Section 8.1) and
RSASSA-PKCS1-v1_5. Then the security proof for RSASSA-PSS would no
longer be sufficient since the proof does not account for the
possibility that signatures might be generated with a second scheme.
Similar considerations may apply if an RSA key pair is employed in
one of the schemes defined here and in a variant defined elsewhere.
7. Encryption Schemes
For the purposes of this document, an encryption scheme consists of
an encryption operation and a decryption operation, where the
encryption operation produces a ciphertext from a message with a
recipient's RSA public key, and the decryption operation recovers the
message from the ciphertext with the recipient's corresponding RSA
private key.
An encryption scheme can be employed in a variety of applications. A
typical application is a key establishment protocol, where the
message contains key material to be delivered confidentially from one
party to another. For instance, PKCS #7 [RFC2315] employs such a
protocol to deliver a content-encryption key from a sender to a
recipient; the encryption schemes defined here would be suitable key-
encryption algorithms in that context.
Two encryption schemes are specified in this document: RSAES-OAEP and
RSAES-PKCS1-v1_5. RSAES-OAEP is REQUIRED to be supported for new
applications; RSAES-PKCS1-v1_5 is included only for compatibility
with existing applications.
The encryption schemes given here follow a general model similar to
that employed in IEEE 1363 [IEEE1363], combining encryption and
decryption primitives with an encoding method for encryption. The
encryption operations apply a message encoding operation to a message
to produce an encoded message, which is then converted to an integer
message representative. An encryption primitive is applied to the
message representative to produce the ciphertext. Reversing this,
the decryption operations apply a decryption primitive to the
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RFC 8017 PKCS #1 v2.2 November 2016
ciphertext to recover a message representative, which is then
converted to an octet-string-encoded message. A message decoding
operation is applied to the encoded message to recover the message
and verify the correctness of the decryption.
To avoid implementation weaknesses related to the way errors are
handled within the decoding operation (see [BLEICHENBACHER] and
[MANGER]), the encoding and decoding operations for RSAES-OAEP and
RSAES-PKCS1-v1_5 are embedded in the specifications of the respective
encryption schemes rather than defined in separate specifications.
Both encryption schemes are compatible with the corresponding schemes
in PKCS #1 v2.1.
7.1. RSAES-OAEP
RSAES-OAEP combines the RSAEP and RSADP primitives (Sections 5.1.1
and 5.1.2) with the EME-OAEP encoding method (Step 2 in
Section 7.1.1, and Step 3 in Section 7.1.2). EME-OAEP is based on
Bellare and Rogaway's Optimal Asymmetric Encryption scheme [OAEP].
It is compatible with the Integer Factorization Encryption Scheme
(IFES) defined in IEEE 1363 [IEEE1363], where the encryption and
decryption primitives are IFEP-RSA and IFDP-RSA and the message
encoding method is EME-OAEP. RSAES-OAEP can operate on messages of
length up to k - 2hLen -2 octets, where hLen is the length of the
output from the underlying hash function and k is the length in
octets of the recipient's RSA modulus.
Assuming that computing e-th roots modulo n is infeasible and the
mask generation function in RSAES-OAEP has appropriate properties,
RSAES-OAEP is semantically secure against adaptive chosen-ciphertext
attacks. This assurance is provable in the sense that the difficulty
of breaking RSAES-OAEP can be directly related to the difficulty of
inverting the RSA function, provided that the mask generation
function is viewed as a black box or random oracle; see [FOPS] and
the note below for further discussion.
Both the encryption and the decryption operations of RSAES-OAEP take
the value of a label L as input. In this version of PKCS #1, L is
the empty string; other uses of the label are outside the scope of
this document. See Appendix A.2.1 for the relevant ASN.1 syntax.
RSAES-OAEP is parameterized by the choice of hash function and mask
generation function. This choice should be fixed for a given RSA
key. Suggested hash and mask generation functions are given in
Appendix B.
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RFC 8017 PKCS #1 v2.2 November 2016
Note: Past results have helpfully clarified the security properties
of the OAEP encoding method [OAEP] (roughly the procedure described
in Step 2 in Section 7.1.1). The background is as follows. In 1994,
Bellare and Rogaway [OAEP] introduced a security concept that they
denoted plaintext awareness (PA94). They proved that if a
deterministic public-key encryption primitive (e.g., RSAEP) is hard
to invert without the private key, then the corresponding OAEP-based
encryption scheme is plaintext aware (in the random oracle model),
meaning roughly that an adversary cannot produce a valid ciphertext
without actually "knowing" the underlying plaintext. Plaintext
awareness of an encryption scheme is closely related to the
resistance of the scheme against chosen-ciphertext attacks. In such
attacks, an adversary is given the opportunity to send queries to an
oracle simulating the decryption primitive. Using the results of
these queries, the adversary attempts to decrypt a challenge
ciphertext.
However, there are two flavors of chosen-ciphertext attacks, and PA94
implies security against only one of them. The difference relies on
what the adversary is allowed to do after she is given the challenge
ciphertext. The indifferent attack scenario (denoted CCA1) does not
admit any queries to the decryption oracle after the adversary is
given the challenge ciphertext, whereas the adaptive scenario
(denoted CCA2) does (except that the decryption oracle refuses to
decrypt the challenge ciphertext once it is published). In 1998,
Bellare and Rogaway, together with Desai and Pointcheval [PA98], came
up with a new, stronger notion of plaintext awareness (PA98) that
does imply security against CCA2.
To summarize, there have been two potential sources for
misconception: that PA94 and PA98 are equivalent concepts, or that
CCA1 and CCA2 are equivalent concepts. Either assumption leads to
the conclusion that the Bellare-Rogaway paper implies security of
OAEP against CCA2, which it does not.
(Footnote: It might be fair to mention that PKCS #1 v2.0 cites [OAEP]
and claims that "a chosen ciphertext attack is ineffective against a
plaintext-aware encryption scheme such as RSAES-OAEP" without
specifying the kind of plaintext awareness or chosen ciphertext
attack considered.)
OAEP has never been proven secure against CCA2; in fact, Victor Shoup
[SHOUP] has demonstrated that such a proof does not exist in the
general case. Put briefly, Shoup showed that an adversary in the
CCA2 scenario who knows how to partially invert the encryption
primitive but does not know how to invert it completely may well be
able to break the scheme. For example, one may imagine an attacker
who is able to break RSAES-OAEP if she knows how to recover all but
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RFC 8017 PKCS #1 v2.2 November 2016
the first 20 bytes of a random integer encrypted with RSAEP. Such an
attacker does not need to be able to fully invert RSAEP, because she
does not use the first 20 octets in her attack.
Still, RSAES-OAEP is secure against CCA2, which was proved by
Fujisaki, Okamoto, Pointcheval, and Stern [FOPS] shortly after the
announcement of Shoup's result. Using clever lattice reduction
techniques, they managed to show how to invert RSAEP completely given
a sufficiently large part of the pre-image. This observation,
combined with a proof that OAEP is secure against CCA2 if the
underlying encryption primitive is hard to partially invert, fills
the gap between what Bellare and Rogaway proved about RSAES-OAEP and
what some may have believed that they proved. Somewhat
paradoxically, we are hence saved by an ostensible weakness in RSAEP
(i.e., the whole inverse can be deduced from parts of it).
Unfortunately, however, the security reduction is not efficient for
concrete parameters. While the proof successfully relates an
adversary A against the CCA2 security of RSAES-OAEP to an algorithm I
inverting RSA, the probability of success for I is only approximately
\epsilon^2 / 2^18, where \epsilon is the probability of success for
A.
(Footnote: In [FOPS], the probability of success for the inverter was
\epsilon^2 / 4. The additional factor 1 / 2^16 is due to the eight
fixed zero bits at the beginning of the encoded message EM, which are
not present in the variant of OAEP considered in [FOPS]. (A must be
applied twice to invert RSA, and each application corresponds to a
factor 1 / 2^8.))
In addition, the running time for I is approximately t^2, where t is
the running time of the adversary. The consequence is that we cannot
exclude the possibility that attacking RSAES-OAEP is considerably
easier than inverting RSA for concrete parameters. Still, the
existence of a security proof provides some assurance that the
RSAES-OAEP construction is sounder than ad hoc constructions such as
RSAES-PKCS1-v1_5.
Hybrid encryption schemes based on the RSA Key Encapsulation
Mechanism (RSA-KEM) paradigm offer tight proofs of security directly
applicable to concrete parameters; see [ISO18033] for discussion.
Future versions of PKCS #1 may specify schemes based on this
paradigm.
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4. Output the message M.
Note: Care must be taken to ensure that an opponent cannot
distinguish the different error conditions in Step 3.g, whether by
error message or timing, and, more generally, that an opponent
cannot learn partial information about the encoded message EM.
Otherwise, an opponent may be able to obtain useful information
about the decryption of the ciphertext C, leading to a chosen-
ciphertext attack such as the one observed by Manger [MANGER].
7.2. RSAES-PKCS1-v1_5
RSAES-PKCS1-v1_5 combines the RSAEP and RSADP primitives (Sections
5.1.1 and 5.1.2) with the EME-PKCS1-v1_5 encoding method (Step 2 in
Section 7.2.1, and Step 3 in Section 7.2.2). It is mathematically
equivalent to the encryption scheme in PKCS #1 v1.5.
RSAES-PKCS1-v1_5 can operate on messages of length up to k - 11
octets (k is the octet length of the RSA modulus), although care
should be taken to avoid certain attacks on low-exponent RSA due to
Coppersmith, Franklin, Patarin, and Reiter when long messages are
encrypted (see the third bullet in the notes below and [LOWEXP];
[NEWATTACK] contains an improved attack). As a general rule, the use
of this scheme for encrypting an arbitrary message, as opposed to a
randomly generated key, is NOT RECOMMENDED.
It is possible to generate valid RSAES-PKCS1-v1_5 ciphertexts without
knowing the corresponding plaintexts, with a reasonable probability
of success. This ability can be exploited in a chosen-ciphertext
attack as shown in [BLEICHENBACHER]. Therefore, if RSAES-PKCS1-v1_5
is to be used, certain easily implemented countermeasures should be
taken to thwart the attack found in [BLEICHENBACHER]. Typical
examples include the addition of structure to the data to be encoded,
rigorous checking of PKCS #1 v1.5 conformance (and other redundancy)
in decrypted messages, and the consolidation of error messages in a
client-server protocol based on PKCS #1 v1.5. These can all be
effective countermeasures and do not involve changes to a protocol
based on PKCS #1 v1.5. See [BKS] for a further discussion of these
and other countermeasures. It has recently been shown that the
security of the SSL/TLS handshake protocol [RFC5246], which uses
RSAES-PKCS1-v1_5 and certain countermeasures, can be related to a
variant of the RSA problem; see [RSATLS] for discussion.
Note: The following passages describe some security recommendations
pertaining to the use of RSAES-PKCS1-v1_5. Recommendations from PKCS
#1 v1.5 are included as well as new recommendations motivated by
cryptanalytic advances made in the intervening years.
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RFC 8017 PKCS #1 v2.2 November 2016
o It is RECOMMENDED that the pseudorandom octets in Step 2 in
Section 7.2.1 be generated independently for each encryption
process, especially if the same data is input to more than one
encryption process. Haastad's results [HAASTAD] are one
motivation for this recommendation.
o The padding string PS in Step 2 in Section 7.2.1 is at least eight
octets long, which is a security condition for public-key
operations that makes it difficult for an attacker to recover data
by trying all possible encryption blocks.
o The pseudorandom octets can also help thwart an attack due to
Coppersmith et al. [LOWEXP] (see [NEWATTACK] for an improvement
of the attack) when the size of the message to be encrypted is
kept small. The attack works on low-exponent RSA when similar
messages are encrypted with the same RSA public key. More
specifically, in one flavor of the attack, when two inputs to
RSAEP agree on a large fraction of bits (8/9) and low-exponent RSA
(e = 3) is used to encrypt both of them, it may be possible to
recover both inputs with the attack. Another flavor of the attack
is successful in decrypting a single ciphertext when a large
fraction (2/3) of the input to RSAEP is already known. For
typical applications, the message to be encrypted is short (e.g.,
a 128-bit symmetric key), so not enough information will be known
or common between two messages to enable the attack. However, if
a long message is encrypted, or if part of a message is known,
then the attack may be a concern. In any case, the RSAES-OAEP
scheme overcomes the attack.
7.2.1. Encryption Operation
RSAES-PKCS1-V1_5-ENCRYPT ((n, e), M)
Input:
(n, e) recipient's RSA public key (k denotes the length in
octets of the modulus n)
M message to be encrypted, an octet string of length
mLen, where mLen <= k - 11
Output:
C ciphertext, an octet string of length k
Error: "message too long"
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Note: Care shall be taken to ensure that an opponent cannot
distinguish the different error conditions in Step 3, whether by
error message or timing. Otherwise, an opponent may be able to
obtain useful information about the decryption of the ciphertext
C, leading to a strengthened version of Bleichenbacher's attack
[BLEICHENBACHER]; compare to Manger's attack [MANGER].
8. Signature Scheme with Appendix
For the purposes of this document, a signature scheme with appendix
consists of a signature generation operation and a signature
verification operation, where the signature generation operation
produces a signature from a message with a signer's RSA private key,
and the signature verification operation verifies the signature on
the message with the signer's corresponding RSA public key. To
verify a signature constructed with this type of scheme, it is
necessary to have the message itself. In this way, signature schemes
with appendix are distinguished from signature schemes with message
recovery, which are not supported in this document.
A signature scheme with appendix can be employed in a variety of
applications. For instance, the signature schemes with appendix
defined here would be suitable signature algorithms for X.509
certificates [ISO9594]. Related signature schemes could be employed
in PKCS #7 [RFC2315], although for technical reasons the current
version of PKCS #7 separates a hash function from a signature scheme,
which is different than what is done here; see the note in
Appendix A.2.3 for more discussion.
Two signature schemes with appendix are specified in this document:
RSASSA-PSS and RSASSA-PKCS1-v1_5. Although no attacks are known
against RSASSA-PKCS1-v1_5, in the interest of increased robustness,
RSASSA-PSS is REQUIRED in new applications. RSASSA-PKCS1-v1_5 is
included only for compatibility with existing applications.
The signature schemes with appendix given here follow a general model
similar to that employed in IEEE 1363 [IEEE1363], combining signature
and verification primitives with an encoding method for signatures.
The signature generation operations apply a message encoding
operation to a message to produce an encoded message, which is then
converted to an integer message representative. A signature
primitive is applied to the message representative to produce the
signature. Reversing this, the signature verification operations
apply a signature verification primitive to the signature to recover
a message representative, which is then converted to an octet-string-
encoded message. A verification operation is applied to the message
and the encoded message to determine whether they are consistent.
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If the encoding method is deterministic (e.g., EMSA-PKCS1-v1_5), the
verification operation may apply the message encoding operation to
the message and compare the resulting encoded message to the
previously derived encoded message. If there is a match, the
signature is considered valid. If the method is randomized (e.g.,
EMSA-PSS), the verification operation is typically more complicated.
For example, the verification operation in EMSA-PSS extracts the
random salt and a hash output from the encoded message and checks
whether the hash output, the salt, and the message are consistent;
the hash output is a deterministic function in terms of the message
and the salt. For both signature schemes with appendix defined in
this document, the signature generation and signature verification
operations are readily implemented as "single-pass" operations if the
signature is placed after the message. See PKCS #7 [RFC2315] for an
example format in the case of RSASSA-PKCS1-v1_5.
8.1. RSASSA-PSS
RSASSA-PSS combines the RSASP1 and RSAVP1 primitives with the
EMSA-PSS encoding method. It is compatible with the Integer
Factorization Signature Scheme with Appendix (IFSSA) as amended in
IEEE 1363a [IEEE1363A], where the signature and verification
primitives are IFSP-RSA1 and IFVP-RSA1 as defined in IEEE 1363
[IEEE1363], and the message encoding method is EMSA4. EMSA4 is
slightly more general than EMSA-PSS as it acts on bit strings rather
than on octet strings. EMSA-PSS is equivalent to EMSA4 restricted to
the case that the operands as well as the hash and salt values are
octet strings.
The length of messages on which RSASSA-PSS can operate is either
unrestricted or constrained by a very large number, depending on the
hash function underlying the EMSA-PSS encoding method.
Assuming that computing e-th roots modulo n is infeasible and the
hash and mask generation functions in EMSA-PSS have appropriate
properties, RSASSA-PSS provides secure signatures. This assurance is
provable in the sense that the difficulty of forging signatures can
be directly related to the difficulty of inverting the RSA function,
provided that the hash and mask generation functions are viewed as
black boxes or random oracles. The bounds in the security proof are
essentially "tight", meaning that the success probability and running
time for the best forger against RSASSA-PSS are very close to the
corresponding parameters for the best RSA inversion algorithm; see
[RSARABIN] [PSSPROOF] [JONSSON] for further discussion.
In contrast to the RSASSA-PKCS1-v1_5 signature scheme, a hash
function identifier is not embedded in the EMSA-PSS encoded message,
so in theory it is possible for an adversary to substitute a
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different (and potentially weaker) hash function than the one
selected by the signer. Therefore, it is RECOMMENDED that the
EMSA-PSS mask generation function be based on the same hash function.
In this manner, the entire encoded message will be dependent on the
hash function, and it will be difficult for an opponent to substitute
a different hash function than the one intended by the signer. This
matching of hash functions is only for the purpose of preventing hash
function substitution and is not necessary if hash function
substitution is addressed by other means (e.g., the verifier accepts
only a designated hash function). See [HASHID] for further
discussion of these points. The provable security of RSASSA-PSS does
not rely on the hash function in the mask generation function being
the same as the hash function applied to the message.
RSASSA-PSS is different from other RSA-based signature schemes in
that it is probabilistic rather than deterministic, incorporating a
randomly generated salt value. The salt value enhances the security
of the scheme by affording a "tighter" security proof than
deterministic alternatives such as Full Domain Hashing (FDH); see
[RSARABIN] for discussion. However, the randomness is not critical
to security. In situations where random generation is not possible,
a fixed value or a sequence number could be employed instead, with
the resulting provable security similar to that of FDH [FDH].
8.1.1. Signature Generation Operation
RSASSA-PSS-SIGN (K, M)
Input:
K signer's RSA private key
M message to be signed, an octet string
Output:
S signature, an octet string of length k, where k is the
length in octets of the RSA modulus n
Errors: "message too long;" "encoding error"
Steps:
1. EMSA-PSS encoding: Apply the EMSA-PSS encoding operation
(Section 9.1.1) to the message M to produce an encoded message
EM of length \ceil ((modBits - 1)/8) octets such that the bit
length of the integer OS2IP (EM) (see Section 4.2) is at most
modBits - 1, where modBits is the length in bits of the RSA
modulus n:
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2. RSA verification:
a. Convert the signature S to an integer signature
representative s (see Section 4.2):
s = OS2IP (S).
b. Apply the RSAVP1 verification primitive (Section 5.2.2) to
the RSA public key (n, e) and the signature representative
s to produce an integer message representative m:
m = RSAVP1 ((n, e), s).
If RSAVP1 output "signature representative out of range",
output "invalid signature" and stop.
c. Convert the message representative m to an encoded message
EM of length emLen = \ceil ((modBits - 1)/8) octets, where
modBits is the length in bits of the RSA modulus n (see
Section 4.1):
EM = I2OSP (m, emLen).
Note that emLen will be one less than k if modBits - 1 is
divisible by 8 and equal to k otherwise. If I2OSP outputs
"integer too large", output "invalid signature" and stop.
3. EMSA-PSS verification: Apply the EMSA-PSS verification
operation (Section 9.1.2) to the message M and the encoded
message EM to determine whether they are consistent:
Result = EMSA-PSS-VERIFY (M, EM, modBits - 1).
4. If Result = "consistent", output "valid signature".
Otherwise, output "invalid signature".
8.2. RSASSA-PKCS1-v1_5
RSASSA-PKCS1-v1_5 combines the RSASP1 and RSAVP1 primitives with the
EMSA-PKCS1-v1_5 encoding method. It is compatible with the IFSSA
scheme defined in IEEE 1363 [IEEE1363], where the signature and
verification primitives are IFSP-RSA1 and IFVP-RSA1, and the message
encoding method is EMSA-PKCS1-v1_5 (which is not defined in IEEE 1363
but is in IEEE 1363a [IEEE1363A]).
The length of messages on which RSASSA-PKCS1-v1_5 can operate is
either unrestricted or constrained by a very large number, depending
on the hash function underlying the EMSA-PKCS1-v1_5 method.
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Assuming that computing e-th roots modulo n is infeasible and the
hash function in EMSA-PKCS1-v1_5 has appropriate properties,
RSASSA-PKCS1-v1_5 is conjectured to provide secure signatures. More
precisely, forging signatures without knowing the RSA private key is
conjectured to be computationally infeasible. Also, in the encoding
method EMSA-PKCS1-v1_5, a hash function identifier is embedded in the
encoding. Because of this feature, an adversary trying to find a
message with the same signature as a previously signed message must
find collisions of the particular hash function being used; attacking
a different hash function than the one selected by the signer is not
useful to the adversary. See [HASHID] for further discussion.
Note: As noted in PKCS #1 v1.5, the EMSA-PKCS1-v1_5 encoding method
has the property that the encoded message, converted to an integer
message representative, is guaranteed to be large and at least
somewhat "random". This prevents attacks of the kind proposed by
Desmedt and Odlyzko [CHOSEN] where multiplicative relationships
between message representatives are developed by factoring the
message representatives into a set of small values (e.g., a set of
small primes). Coron, Naccache, and Stern [PADDING] showed that a
stronger form of this type of attack could be quite effective against
some instances of the ISO/IEC 9796-2 signature scheme. They also
analyzed the complexity of this type of attack against the
EMSA-PKCS1-v1_5 encoding method and concluded that an attack would be
impractical, requiring more operations than a collision search on the
underlying hash function (i.e., more than 2^80 operations).
Coppersmith, Halevi, and Jutla [FORGERY] subsequently extended Coron
et al.'s attack to break the ISO/IEC 9796-1 signature scheme with
message recovery. The various attacks illustrate the importance of
carefully constructing the input to the RSA signature primitive,
particularly in a signature scheme with message recovery.
Accordingly, the EMSA-PKCS-v1_5 encoding method explicitly includes a
hash operation and is not intended for signature schemes with message
recovery. Moreover, while no attack is known against the
EMSA-PKCS-v1_5 encoding method, a gradual transition to EMSA-PSS is
recommended as a precaution against future developments.
8.2.1. Signature Generation Operation
RSASSA-PKCS1-V1_5-SIGN (K, M)
Input:
K signer's RSA private key
M message to be signed, an octet string
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4. Compare the encoded message EM and the second encoded message
EM'. If they are the same, output "valid signature";
otherwise, output "invalid signature".
Note: Another way to implement the signature verification
operation is to apply a "decoding" operation (not specified in
this document) to the encoded message to recover the underlying
hash value, and then compare it to a newly computed hash value.
This has the advantage that it requires less intermediate storage
(two hash values rather than two encoded messages), but the
disadvantage that it requires additional code.
9. Encoding Methods for Signatures with Appendix
Encoding methods consist of operations that map between octet string
messages and octet-string-encoded messages, which are converted to
and from integer message representatives in the schemes. The integer
message representatives are processed via the primitives. The
encoding methods thus provide the connection between the schemes,
which process messages, and the primitives.
An encoding method for signatures with appendix, for the purposes of
this document, consists of an encoding operation and optionally a
verification operation. An encoding operation maps a message M to an
encoded message EM of a specified length. A verification operation
determines whether a message M and an encoded message EM are
consistent, i.e., whether the encoded message EM is a valid encoding
of the message M.
The encoding operation may introduce some randomness, so that
different applications of the encoding operation to the same message
will produce different encoded messages, which has benefits for
provable security. For such an encoding method, both an encoding and
a verification operation are needed unless the verifier can reproduce
the randomness (e.g., by obtaining the salt value from the signer).
For a deterministic encoding method, only an encoding operation is
needed.
Two encoding methods for signatures with appendix are employed in the
signature schemes and are specified here: EMSA-PSS and
EMSA-PKCS1-v1_5.
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Notes:
1. The encoding method defined here differs from the one in Bellare
and Rogaway's submission to IEEE 1363a [PSS] in three respects:
* It applies a hash function rather than a mask generation
function to the message. Even though the mask generation
function is based on a hash function, it seems more natural to
apply a hash function directly.
* The value that is hashed together with the salt value is the
string (0x)00 00 00 00 00 00 00 00 || mHash rather than the
message M itself. Here, mHash is the hash of M. Note that
the hash function is the same in both steps. See Note 3 below
for further discussion. (Also, the name "salt" is used
instead of "seed", as it is more reflective of the value's
role.)
* The encoded message in EMSA-PSS has nine fixed bits; the first
bit is 0 and the last eight bits form a "trailer field", the
octet 0xbc. In the original scheme, only the first bit is
fixed. The rationale for the trailer field is for
compatibility with the Integer Factorization Signature
Primitive using Rabin-Williams (IFSP-RW) in IEEE 1363
[IEEE1363] and the corresponding primitive in ISO/IEC
9796-2:2010 [ISO9796].
2. Assuming that the mask generation function is based on a hash
function, it is RECOMMENDED that the hash function be the same as
the one that is applied to the message; see Section 8.1 for
further discussion.
3. Without compromising the security proof for RSASSA-PSS, one may
perform Steps 1 and 2 of EMSA-PSS-ENCODE and EMSA-PSS-VERIFY (the
application of the hash function to the message) outside the
module that computes the rest of the signature operation, so that
mHash rather than the message M itself is input to the module.
In other words, the security proof for RSASSA-PSS still holds
even if an opponent can control the value of mHash. This is
convenient if the module has limited I/O bandwidth, e.g., a smart
card. Note that previous versions of PSS [RSARABIN][PSS] did not
have this property. Of course, it may be desirable for other
security reasons to have the module process the full message.
For instance, the module may need to "see" what it is signing if
it does not trust the component that computes the hash value.
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4. Typical salt lengths in octets are hLen (the length of the output
of the hash function Hash) and 0. In both cases, the security of
RSASSA-PSS can be closely related to the hardness of inverting
RSAVP1. Bellare and Rogaway [RSARABIN] give a tight lower bound
for the security of the original RSA-PSS scheme, which
corresponds roughly to the former case, while Coron [FDH] gives a
lower bound for the related Full Domain Hashing scheme, which
corresponds roughly to the latter case. In [PSSPROOF], Coron
provides a general treatment with various salt lengths ranging
from 0 to hLen; see [IEEE1363A] for discussion. See also
[JONSSON], which adapts the security proofs in [RSARABIN]
[PSSPROOF] to address the differences between the original and
the present version of RSA-PSS as listed in Note 1 above.
5. As noted in IEEE 1363a [IEEE1363A], the use of randomization in
signature schemes -- such as the salt value in EMSA-PSS -- may
provide a "covert channel" for transmitting information other
than the message being signed. For more on covert channels, see
[SIMMONS].
9.1.1. Encoding Operation
EMSA-PSS-ENCODE (M, emBits)
Options:
Hash hash function (hLen denotes the length in octets of
the hash function output)
MGF mask generation function
sLen intended length in octets of the salt
Input:
M message to be encoded, an octet string
emBits maximal bit length of the integer OS2IP (EM) (see Section4.2), at least 8hLen + 8sLen + 9
Output:
EM encoded message, an octet string of length emLen = \ceil
(emBits/8)
Errors: "Encoding error"; "message too long"
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RFC 8017 PKCS #1 v2.2 November 2016Appendix A. ASN.1 SyntaxA.1. RSA Key Representation
This section defines ASN.1 object identifiers for RSA public and
private keys and defines the types RSAPublicKey and RSAPrivateKey.
The intended application of these definitions includes X.509
certificates, PKCS #8 [RFC5958], and PKCS #12 [RFC7292].
The object identifier rsaEncryption identifies RSA public and private
keys as defined in Appendices A.1.1 and A.1.2. The parameters field
has associated with this OID in a value of type AlgorithmIdentifier
SHALL have a value of type NULL.
rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 }
The definitions in this section have been extended to support multi-
prime RSA, but they are backward compatible with previous versions.
A.1.1. RSA Public Key Syntax
An RSA public key should be represented with the ASN.1 type
RSAPublicKey:
RSAPublicKey ::= SEQUENCE {
modulus INTEGER, -- n
publicExponent INTEGER -- e
}
The fields of type RSAPublicKey have the following meanings:
o modulus is the RSA modulus n.
o publicExponent is the RSA public exponent e.
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o otherPrimeInfos contains the information for the additional primes
r_3, ..., r_u, in order. It SHALL be omitted if version is 0 and
SHALL contain at least one instance of OtherPrimeInfo if version
is 1.
OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo
OtherPrimeInfo ::= SEQUENCE {
prime INTEGER, -- ri
exponent INTEGER, -- di
coefficient INTEGER -- ti
}
The fields of type OtherPrimeInfo have the following meanings:
o prime is a prime factor r_i of n, where i >= 3.
o exponent is d_i = d mod (r_i - 1).
o coefficient is the CRT coefficient t_i = (r_1 * r_2 * ... *
r_(i-1))^(-1) mod r_i.
Note: It is important to protect the RSA private key against both
disclosure and modification. Techniques for such protection are
outside the scope of this document. Methods for storing and
distributing private keys and other cryptographic data are described
in PKCS #12 and #15.
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o pSourceAlgorithm identifies the source (and possibly the value) of
the label L. It SHALL be an algorithm ID with an OID in the set
PKCS1PSourceAlgorithms, which for this version SHALL consist of
id-pSpecified, indicating that the label is specified explicitly.
The parameters field associated with id-pSpecified SHALL have a
value of type OCTET STRING, containing the label. In previous
versions of this specification, the term "encoding parameters" was
used rather than "label", hence the name of the type below.
PSourceAlgorithm ::= AlgorithmIdentifier {
{PKCS1PSourceAlgorithms}
}
PKCS1PSourceAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-pSpecified PARAMETERS EncodingParameters },
... -- Allows for future expansion --
}
id-pSpecified OBJECT IDENTIFIER ::= { pkcs-1 9 }
EncodingParameters ::= OCTET STRING(SIZE(0..MAX))
o The default label is an empty string (so that lHash will contain
the hash of the empty string):
pSpecifiedEmpty PSourceAlgorithm ::= {
algorithm id-pSpecified,
parameters EncodingParameters : emptyString
}
emptyString EncodingParameters ::= ''H
If all of the default values of the fields in RSAES-OAEP-params are
used, then the algorithm identifier will have the following value:
rSAES-OAEP-Default-Identifier RSAES-AlgorithmIdentifier ::= {
algorithm id-RSAES-OAEP,
parameters RSAES-OAEP-params : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
pSourceAlgorithm pSpecifiedEmpty
}
}
RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier {
{PKCS1Algorithms}
}
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RFC 8017 PKCS #1 v2.2 November 2016A.2.2. RSAES-PKCS-v1_5
The object identifier rsaEncryption (see Appendix A.1) identifies the
RSAES-PKCS1-v1_5 encryption scheme. The parameters field associated
with this OID in a value of type AlgorithmIdentifier SHALL have a
value of type NULL. This is the same as in PKCS #1 v1.5.
rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 }
A.2.3. RSASSA-PSS
The object identifier id-RSASSA-PSS identifies the RSASSA-PSS
encryption scheme.
id-RSASSA-PSS OBJECT IDENTIFIER ::= { pkcs-1 10 }
The parameters field associated with this OID in a value of type
AlgorithmIdentifier SHALL have a value of type RSASSA-PSS-params:
RSASSA-PSS-params ::= SEQUENCE {
hashAlgorithm [0] HashAlgorithm DEFAULT sha1,
maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1,
saltLength [2] INTEGER DEFAULT 20,
trailerField [3] TrailerField DEFAULT trailerFieldBC
}
The fields of type RSASSA-PSS-params have the following meanings:
o hashAlgorithm identifies the hash function. It SHALL be an
algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms (see
Appendix A.2.1). The default hash function is SHA-1.
o maskGenAlgorithm identifies the mask generation function. It
SHALL be an algorithm ID with an OID in the set PKCS1MGFAlgorithms
(see Appendix A.2.1). The default mask generation function is
MGF1 with SHA-1. For MGF1 (and more generally, for other mask
generation functions based on a hash function), it is RECOMMENDED
that the underlying hash function be the same as the one
identified by hashAlgorithm; see Note 2 in Section 9.1 for further
comments.
o saltLength is the octet length of the salt. It SHALL be an
integer. For a given hashAlgorithm, the default value of
saltLength is the octet length of the hash value. Unlike the
other fields of type RSASSA-PSS-params, saltLength does not need
to be fixed for a given RSA key pair.
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o trailerField is the trailer field number, for compatibility with
IEEE 1363a [IEEE1363A]. It SHALL be 1 for this version of the
document, which represents the trailer field with hexadecimal
value 0xbc. Other trailer fields (including the trailer field
HashID || 0xcc in IEEE 1363a) are not supported in this document.
TrailerField ::= INTEGER { trailerFieldBC(1) }
If the default values of the hashAlgorithm, maskGenAlgorithm, and
trailerField fields of RSASSA-PSS-params are used, then the algorithm
identifier will have the following value:
rSASSA-PSS-Default-Identifier RSASSA-AlgorithmIdentifier ::= {
algorithm id-RSASSA-PSS,
parameters RSASSA-PSS-params : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
saltLength 20,
trailerField trailerFieldBC
}
}
RSASSA-AlgorithmIdentifier ::= AlgorithmIdentifier {
{PKCS1Algorithms}
}
Note: In some applications, the hash function underlying a signature
scheme is identified separately from the rest of the operations in
the signature scheme. For instance, in PKCS #7 [RFC2315], a hash
function identifier is placed before the message and a "digest
encryption" algorithm identifier (indicating the rest of the
operations) is carried with the signature. In order for PKCS #7 to
support the RSASSA-PSS signature scheme, an object identifier would
need to be defined for the operations in RSASSA-PSS after the hash
function (analogous to the RSAEncryption OID for the
RSASSA-PKCS1-v1_5 scheme). S/MIME Cryptographic Message Syntax (CMS)
[RFC5652] takes a different approach. Although a hash function
identifier is placed before the message, an algorithm identifier for
the full signature scheme may be carried with a CMS signature (this
is done for DSA signatures). Following this convention, the
id-RSASSA-PSS OID can be used to identify RSASSA-PSS signatures in
CMS. Since CMS is considered the successor to PKCS #7 and new
developments such as the addition of support for RSASSA-PSS will be
pursued with respect to CMS rather than PKCS #7, an OID for the "rest
of" RSASSA-PSS is not defined in this version of PKCS #1.
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RFC 8017 PKCS #1 v2.2 November 2016A.2.4. RSASSA-PKCS-v1_5
The object identifier for RSASSA-PKCS1-v1_5 SHALL be one of the
following. The choice of OID depends on the choice of hash
algorithm: MD2, MD5, SHA-1, SHA-224, SHA-256, SHA-384, SHA-512,
SHA-512/224, or SHA-512/256. Note that if either MD2 or MD5 is used,
then the OID is just as in PKCS #1 v1.5. For each OID, the
parameters field associated with this OID in a value of type
AlgorithmIdentifier SHALL have a value of type NULL. The OID should
be chosen in accordance with the following table:
Hash algorithm OID
------------------------------------------------------------
MD2 md2WithRSAEncryption ::= {pkcs-1 2}
MD5 md5WithRSAEncryption ::= {pkcs-1 4}
SHA-1 sha1WithRSAEncryption ::= {pkcs-1 5}
SHA-256 sha224WithRSAEncryption ::= {pkcs-1 14}
SHA-256 sha256WithRSAEncryption ::= {pkcs-1 11}
SHA-384 sha384WithRSAEncryption ::= {pkcs-1 12}
SHA-512 sha512WithRSAEncryption ::= {pkcs-1 13}
SHA-512/224 sha512-224WithRSAEncryption ::= {pkcs-1 15}
SHA-512/256 sha512-256WithRSAEncryption ::= {pkcs-1 16}
The EMSA-PKCS1-v1_5 encoding method includes an ASN.1 value of type
DigestInfo, where the type DigestInfo has the syntax
DigestInfo ::= SEQUENCE {
digestAlgorithm DigestAlgorithm,
digest OCTET STRING
}
digestAlgorithm identifies the hash function and SHALL be an
algorithm ID with an OID in the set PKCS1-v1-5DigestAlgorithms. For
a discussion of supported hash functions, see Appendix B.1.
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DigestAlgorithm ::= AlgorithmIdentifier {
{PKCS1-v1-5DigestAlgorithms}
}
PKCS1-v1-5DigestAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-md2 PARAMETERS NULL }|
{ OID id-md5 PARAMETERS NULL }|
{ OID id-sha1 PARAMETERS NULL }|
{ OID id-sha224 PARAMETERS NULL }|
{ OID id-sha256 PARAMETERS NULL }|
{ OID id-sha384 PARAMETERS NULL }|
{ OID id-sha512 PARAMETERS NULL }|
{ OID id-sha512-224 PARAMETERS NULL }|
{ OID id-sha512-256 PARAMETERS NULL }
}
Appendix B. Supporting Techniques
This section gives several examples of underlying functions
supporting the encryption schemes in Section 7 and the encoding
methods in Section 9. A range of techniques is given here to allow
compatibility with existing applications as well as migration to new
techniques. While these supporting techniques are appropriate for
applications to implement, none of them is required to be
implemented. It is expected that profiles for PKCS #1 v2.2 will be
developed that specify particular supporting techniques.
This section also gives object identifiers for the supporting
techniques.
B.1. Hash Functions
Hash functions are used in the operations contained in Sections 7 and
9. Hash functions are deterministic, meaning that the output is
completely determined by the input. Hash functions take octet
strings of variable length and generate fixed-length octet strings.
The hash functions used in the operations contained in Sections 7 and
9 should generally be collision-resistant. This means that it is
infeasible to find two distinct inputs to the hash function that
produce the same output. A collision-resistant hash function also
has the desirable property of being one-way; this means that given an
output, it is infeasible to find an input whose hash is the specified
output. In addition to the requirements, the hash function should
yield a mask generation function (Appendix B.2) with pseudorandom
output.
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RFC 8017 PKCS #1 v2.2 November 2016
id-sha512 OBJECT IDENTIFIER ::= {
joint-iso-itu-t (2) country (16) us (840) organization (1)
gov (101) csor (3) nistalgorithm (4) hashalgs (2) 3
}
id-sha512-224 OBJECT IDENTIFIER ::= {
joint-iso-itu-t (2) country (16) us (840) organization (1)
gov (101) csor (3) nistalgorithm (4) hashalgs (2) 5
}
id-sha512-256 OBJECT IDENTIFIER ::= {
joint-iso-itu-t (2) country (16) us (840) organization (1)
gov (101) csor (3) nistalgorithm (4) hashalgs (2) 6
}
The parameters field associated with these OIDs in a value of type
AlgorithmIdentifier SHALL have a value of type NULL.
The parameters field associated with id-md2 and id-md5 in a value of
type AlgorithmIdentifier shall have a value of type NULL.
The parameters field associated with id-sha1, id-sha224, id-sha256,
id-sha384, id-sha512, id-sha512/224, and id-sha512/256 should
generally be omitted, but if present, it shall have a value of type
NULL.
This is to align with the definitions originally promulgated by NIST.
For the SHA algorithms, implementations MUST accept
AlgorithmIdentifier values both without parameters and with NULL
parameters.
Exception: When formatting the DigestInfoValue in EMSA-PKCS1-v1_5
(see Section 9.2), the parameters field associated with id-sha1,
id-sha224, id-sha256, id-sha384, id-sha512, id-sha512/224, and
id-sha512/256 shall have a value of type NULL. This is to maintain
compatibility with existing implementations and with the numeric
information values already published for EMSA-PKCS1-v1_5, which are
also reflected in IEEE 1363a [IEEE1363A].
Note: Version 1.5 of PKCS #1 also allowed for the use of MD4 in
signature schemes. The cryptanalysis of MD4 has progressed
significantly in the intervening years. For example, Dobbertin [MD4]
demonstrated how to find collisions for MD4 and that the first two
rounds of MD4 are not one-way [MD4FIRST]. Because of these results
and others (e.g., [MD4LAST]), MD4 is NOT RECOMMENDED.
Further advances have been made in the cryptanalysis of MD2 and MD5,
especially after the findings of Stevens et al. [PREFIX] on chosen-
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RFC 8017 PKCS #1 v2.2 November 2016
prefix collisions on MD5. MD2 and MD5 should be considered
cryptographically broken and removed from existing applications.
This version of the standard supports MD2 and MD5 just for backwards-
compatibility reasons.
There have also been advances in the cryptanalysis of SHA-1.
Particularly, the results of Wang et al. [SHA1CRYPT] (which have
been independently verified by M. Cochran in his analysis [COCHRAN])
on using a differential path to find collisions in SHA-1, which
conclude that the security strength of the SHA-1 hashing algorithm is
significantly reduced. However, this reduction is not significant
enough to warrant the removal of SHA-1 from existing applications,
but its usage is only recommended for backwards-compatibility
reasons.
To address these concerns, only SHA-224, SHA-256, SHA-384, SHA-512,
SHA-512/224, and SHA-512/256 are RECOMMENDED for new applications.
As of today, the best (known) collision attacks against these hash
functions are generic attacks with complexity 2L/2, where L is the
bit length of the hash output. For the signature schemes in this
document, a collision attack is easily translated into a signature
forgery. Therefore, the value L / 2 should be at least equal to the
desired security level in bits of the signature scheme (a security
level of B bits means that the best attack has complexity 2B). The
same rule of thumb can be applied to RSAES-OAEP; it is RECOMMENDED
that the bit length of the seed (which is equal to the bit length of
the hash output) be twice the desired security level in bits.
B.2. Mask Generation Functions
A mask generation function takes an octet string of variable length
and a desired output length as input and outputs an octet string of
the desired length. There may be restrictions on the length of the
input and output octet strings, but such bounds are generally very
large. Mask generation functions are deterministic; the octet string
output is completely determined by the input octet string. The
output of a mask generation function should be pseudorandom: Given
one part of the output but not the input, it should be infeasible to
predict another part of the output. The provable security of
RSAES-OAEP and RSASSA-PSS relies on the random nature of the output
of the mask generation function, which in turn relies on the random
nature of the underlying hash.
One mask generation function is given here: MGF1, which is based on a
hash function. MGF1 coincides with the mask generation functions
defined in IEEE 1363 [IEEE1363] and ANSI X9.44 [ANSIX944]. Future
versions of this document may define other mask generation functions.
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RFC 8017 PKCS #1 v2.2 November 2016Appendix D. Revision History of PKCS #1
Versions 1.0 - 1.5:
Versions 1.0 - 1.3 were distributed to participants in RSA Data
Security, Inc.'s Public-Key Cryptography Standards meetings in
February and March 1991.
Version 1.4 was part of the June 3, 1991 initial public release of
PKCS. Version 1.4 was published as NIST/OSI Implementors'
Workshop document SEC-SIG-91-18.
Version 1.5 incorporated several editorial changes, including
updates to the references and the addition of a revision history.
The following substantive changes were made:
* Section 10: "MD4 with RSA" signature and verification processes
were added.
* Section 11: md4WithRSAEncryption object identifier was added.
Version 1.5 was republished as [RFC2313] (which was later
obsoleted by [RFC2437]).
Version 2.0:
Version 2.0 incorporated major editorial changes in terms of the
document structure and introduced the RSAES-OAEP encryption
scheme. This version continued to support the encryption and
signature processes in version 1.5, although the hash algorithm
MD4 was no longer allowed due to cryptanalytic advances in the
intervening years. Version 2.0 was republished as [RFC2437]
(which was later obsoleted by [RFC3447]).
Version 2.1:
Version 2.1 introduced multi-prime RSA and the RSASSA-PSS
signature scheme with appendix along with several editorial
improvements. This version continued to support the schemes in
version 2.0. Version 2.1 was republished as [RFC3447].
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RFC 8017 PKCS #1 v2.2 November 2016
Version 2.2:
Version 2.2 updates the list of allowed hashing algorithms to
align them with FIPS 180-4 [SHS], therefore adding SHA-224,
SHA-512/224, and SHA-512/256. The following substantive changes
were made:
* Object identifiers for sha224WithRSAEncryption,
sha512-224WithRSAEncryption, and sha512-256WithRSAEncryption
were added.
* This version continues to support the schemes in version 2.1.
Appendix E. About PKCS
The Public-Key Cryptography Standards are specifications produced by
RSA Laboratories in cooperation with secure systems developers
worldwide for the purpose of accelerating the deployment of public-
key cryptography. First published in 1991 as a result of meetings
with a small group of early adopters of public-key technology, the
PKCS documents have become widely referenced and implemented.
Contributions from the PKCS series have become part of many formal
and de facto standards, including ANSI X9 and IEEE P1363 documents,
PKIX, Secure Electronic Transaction (SET), S/MIME, SSL/TLS, and
Wireless Application Protocol (WAP) / WAP Transport Layer Security
(WTLS).
Further development of most PKCS documents occurs through the IETF.
Suggestions for improvement are welcome.
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